Climbing in balls and horoballs

In this subsection, we construct geodesic rays or lines having prescribed penetration properties in a ball or a horoball, while essentially avoiding a family of almost disjoint convex subsets. Let us consider the penetration height and inner projection penetration maps first in horoballs and then in balls. Note that in these cases, if f₀ = ph_C0, then kf0−ph_C0k∞=0 and if f0=ipp_C0, then kf0−ph_C0k∞≤c^′₁(∞) by Section3.1.

Theorem 5.1 Letǫ ∈R^∗₊∪ {∞}, δ, κ ≥ 0; let X be a complete simply connected Riemannian manifold with sectional curvature at most −1 and dimension at least 3;

let ξ0 ∈ X∪∂_∞X; let C₀ be a horoball such that ξ0 ∈/ C0∪∂_∞C0; let f₀ = ph_C0

or f₀ = ipp_C0; let (C_n)_n∈N−{0} be a family of ǫ-convex subsets of X ; for every n∈ N− {0} such that ξ₀ ∈/ C_n∪∂_∞C_n, let f_n : T_ξ¹

0X → [0,+∞] be a κ-penetration map in Cn. If diam(Cn∩Cm) ≤ δ for all n,m in N with n 6= m, then, for every h≥c^′′₁(ǫ, δ,kf₀−ph_C0k∞), there exists a geodesic ray or line γ starting from ξ₀ and entering C₀at time 0, such that f₀(γ)=h and f_n(γ)≤c^′′₁(ǫ, δ,kf0−ph_C0k∞)+κ for every n≥1such that γ ]δ,+∞[

meets Cn.

Proof. Let h≥c^′′₁ =c^′′₁(ǫ, δ,kf₀−ph_C0k∞). In order to apply Proposition4.10, define ǫ₀ =ǫ, δ₀=δ, κ₀ =c^′₁(∞)=2 log(1+√

2) and h^′₀=h₀(ǫ₀, δ₀, κ₀). Recall thatph_C0 and ipp_C0 are κ0-penetration maps for C0 by Lemma3.3. For every n ∈ N− {0} such thatξ0 ∈/ Cn∪∂_∞Cn, let us apply Proposition3.7Case (1) to C=C0, C^′ =Cn, f =f₀, f^′ =ℓ_Cn, h^′ =h^′₀, so that h^min=2 c^′₁(ǫ)+2δ+kf₀−ph_C0k∞ and h^min₀ =2δ. Note that h^min₀ ≤h^′₀, as

h^′₀≥h^′(ǫ,sinh(δ+c1))≥2 sinh(δ+c1)≥2δ ,

by the definitions of h0 and of h^′(·,·). As h ≥ c^′′₁ ≥ h^min by the definition of c^′′₁, Proposition 3.7 Case (1) hence implies that (Cn)n∈N satisfies the Local prescription property (iv). Thus by Proposition4.10, there exists a geodesic ray or line γ starting at ξ₀ such that f0(γ) = h and ℓ_Cn(γ) ≤ h^′₁(ǫ₀, δ₀,h0(ǫ₀, δ₀, κ₀)), which implies that f_n(γ)≤c^′′₁ +κ, for every n≥1 such thatγ(]δ,+∞[) meets C_n. The proof of the corresponding result when C₀ is a ball of radius R≥ǫis the same, using Case (2) of Proposition 3.7 instead of Case (1). This requires h ≤ h^max = 2R−2c^′₁(ǫ)− kf₀−ph_C0k∞. To be nonempty, the following result requires

R≥c^′′₁(ǫ, δ,kf₀−ph_C0k∞)/2+ c^′₁(ǫ)+kf₀−ph_C0k∞/2.

Theorem 5.2 Letǫ >0,δ, κ≥0; let X be a complete simply connected Riemannian manifold with sectional curvature at most−1and dimension at least 3; let C₀be a ball of radius R≥ǫ; letξ₀ ∈(X∪∂_∞X)−C₀; let f₀=ph_C0 or f₀=ipp_C0; let (C_n)_n∈N−{0} be a family ofǫ-convex subsets of X ; for every n∈N−{0}such thatξ0∈/Cn∪∂_∞Cn, let f_n : T_ξ¹

0X →[0,+∞] be aκ-penetration map in C_n. If diam(C_n∩C_m)≤δ for all n,minNwith n6=m, then, for every

h∈h

c^′′₁ ǫ, δ,kf0−ph_C0k_∞

, 2R−2c^′₁(ǫ)− kf0−ph_C0k_∞i ,

there exists a geodesic ray or line γ starting from ξ0 and entering C0 at time 0, such that f₀(γ) = h and f_n(γ) ≤ c^′′₁ ǫ, δ,kf₀−ph_C0k∞

+κ for every n ≥ 1 such that

γ(]δ,+∞[)meets C_n.

Varying the family (C_n)_n∈N−{0} of ǫ-convex subsets appearing in Theorems 5.1 and 5.2, among balls of radius at least ǫ, horoballs, ǫ-neighbourhoods of totally geodesic subspaces, etc, we get several corollaries. We will only state two of them, Corollaries 5.3and 5.5, which have applications to equivariant families. The proofs of these re-sults are simplified versions of the proof of Corollary4.11, giving better (though very probably not optimal) constants.

Corollary 5.3 Let X be a complete simply connected Riemannian manifold with sec-tional curvature at most−1 and dimension at least 3, and let H_n

n∈N be a family of horoballs in X with pairwise disjoint interiors. Then, for every h ≥ c^′′₁(∞,0,0) ≈ 6.5032, there exists a geodesic line γ^′ such that ph_H0(γ^′) = h and ph_Hn(γ^′) ≤ c^′′₂(∞)≈8.3712 for every n≥1.

Proof. Let c^′′₁ = c^′′₁(∞,0,0) and c^′′₂ =c^′′₂(∞). Let C0 = H0 and let ξ be a point in

∂_∞X−∂_∞C₀. We apply Theorem5.1with ǫ=∞,δ =0, κ=0, ξ₀ =ξ, C_n=H_n for every n inN, f₀ =ph_C0, and f_n=ℓ_Cn for every n6=0 such thatξ₀∈/ C_n∪∂_∞C_n. Note that for every n∈N− {0}, the map fnis aκ-penetration map in Cn. As h≥c^′′₁, there exists a geodesic line γ starting from ξ and entering C₀ at time 0, such that ph_C0(γ) = h and ℓ_Cn(γ) ≤ c^′′₁ for every n ∈ N− {0} such that γ meets C_n at a positive time.

Letξ^′ be the other endpoint ofγ. This point is not in ∂_∞C₀. Applying Theorem5.1 again, as above except that now ξ₀ = ξ^′, we get that there exists a geodesic line γ^′ starting from ξ^′ and entering C0 at time 0, such that ph_C0(γ^′) =h andℓCn(γ^′) ≤c^′′₁ for every n∈N− {0} such thatγ^′ meets C_n at a positive time.

Assume by absurd that there exists n∈N− {0} such that ph_Cn(γ^′) >c^′′₂ >0. Then γ^′ enters Cn at the point x^′_n, exiting it at the point y^′_nat a nonpositive time, as c^′′₂ >c^′′₁. In particular, if x^′ = γ^′(0) is the entering point of γ^′ in C₀, then x^′,y^′_n,x^′_n, ξ^′ are in this order onγ^′ (see the picture in the proof of Corollary4.11).

Let y be the exiting point ofγ out of H0. With c1 =1/19, as in the proof of Lemma 4.6, sinceph_C0(γ) andph_C0(γ^′) are equal to

h≥c^′′₁ ≥h^′₁(∞,0,h₀(∞,0,c^′₁(∞)))≥h₀(∞,0,c^′₁(∞))=h^′(∞,sinh c₁) by the definition of c^′′₁,h^′₁,h0, we have d(x^′,y)≤c1.

Letξ_nbe the point at infinity of H_n. Let p^′ be the point in [x^′_n,y^′_n] the closest to ξ_n, so that

d(p^′,y^′_n)≥β_ξn(y^′_n,p^′)=ph_Cn(γ^′)/2>c^′′₂/2.

Let p be the point of γ the closest to p^′. By convexity and the definition of c^′′₂, we have

d(p^′,p)=d(p^′, γ) ≤d(x^′, γ)≤d(x^′,y)≤c1<c^′′₂/2.

Hence p belongs to the interior of Cn. If p ∈]ξ,y], then the closest point to x^′ on γ lies in ]ξ,p[ and by convexity,

c^′′₂/2<d(p^′,y^′_n)≤d(p^′,x^′)≤d(p^′,p)+d(p,x^′)≤d(p^′,p)+d(y,x^′)≤2c1, a contradiction, as by the definition of c^′′₂, of c^′′₁ and of h^′(ǫ, η) (see the equations (- 5 -) and (- 11 -)), we have

c^′′₂ ≥c^′′₁+2 c₁≥h^′(ǫ,sinh c₁)+2 c₁ ≥2 sinh c₁+2c₁ >4 c₁ .

Hence p ∈ ]y, ξ^′[ ⊂ γ(]0,+∞[), so that γ meets Cn at a positive time. But, by Lemma3.3and the definition of c^′′₂,

ℓ_Cn(γ)≥ph_Cn(γ)−c^′₁(∞) ≥2β_ξn(y^′_n,p)−c^′₁(∞)

≥2(β_ξn(y^′_n,p^′)−d(p,p^′))−c^′₁(∞)>2(c^′′₂/2−c₁)−c^′₁(∞)=c^′′₁ .

This contradicts the construction ofγ.

Let M be a complete nonelementary geometrically finite Riemannian manifold with sectional curvature at most−1 (see for instance [Bow] for a general reference). Recall that a cusp of M is an asymptotic class of minimizing geodesic rays in M along which the injectivity radius converges to 0. If M has finite volume, then the set of cusps of M is in bijection with the (finite) set of ends of M , by the map which associates to a representative of a cusp the end of M towards which it converges. Let π : Me → M be a universal Riemannian covering of M , with covering group Γ. If e is a cusp of M , andρe a minimizing geodesic ray in the class e, as M is geometrically finite and nonelementary, there exists a horoball H_e inM centered at the point at infinitye ξ_e of a fixed liftρe_e ofρ_e inM , such thate γH_e and He have disjoint interiors ifγ ∈Γdoes not fix ξe (see for instance [BK,Bow,HP5]). This horoball is unique if maximal (for the inclusion). The image V_e of H_e in M is called a Margulis neighbourhood of e, and the maximal Margulis neighbourhood of e if He is maximal. In this Section5, we assume that He is maximal, and thatρee starts from the boundary of He. Let hte : M →R be the map defined by

ht_e(x)= lim

t→∞(t−d(ρ_e(t),x)),

called the height function with respect to e. Let maxht_e : T¹M→Rbe defined by maxht_e(γ)=sup

t∈R

ht_e(γ(t)).

The maximum height spectrum of the pair (M,e) is the subset of ]− ∞,+∞] defined by

MaxSp(M,e)=maxht_e(T¹M).

Corollary 5.4 Let M be a complete, nonelementary geometrically finite Riemannian manifold with sectional curvature at most−1and dimension at least 3, and let e be a cusp of M . Then MaxSp(M,e)contains [c^′′₂(∞)/2,+∞].

Note that c^′′₂(∞)/2≈4.1856, hence Theorem1.2of the introduction follows.

Proof. With the above notations, let (H_n)_n∈N be the Γ-equivariant family of horoballs in M with pairwise disjoint interiors such that He 0 = He. Apply Corollary 5.3 to this family to get, for every h ≥ c^′′₂(∞) ≥ c^′′₁(∞,0,0), a geodesic line γe in M withe ph_H0(γe) = h and ph_Hn(eγ) ≤ c^′′₂(∞) for every n ≥ 1. Let γ be the locally geodesic line in M image by π of eγ. Observe that hte◦π = βHn in Hn and that ph_Hn(eγ) = 2 sup_t∈Rβ_Hn(eγ(t)) (see Section 3.1). Hence sup_t∈Rht_e(γ(t)) = h/2 and the result

follows.

Schmidt and Sheingorn [SS] treated the case of two-dimensional manifolds of con-stant curvature −1 (hyperbolic surfaces) with a cusp. They showed that in that case MaxSp(M,e) contains the interval [log 100,+∞] ≈ [4.61,+∞]. This paper [SS]

was a starting point of our investigations, although the method we use is quite differ-ent from theirs.

Let P be a (necessarily finite) nonempty set of cusps of M . For every e in P, choose a maximal horoball He, with point at infinity ξe as above Corollary 5.4. The horoballs of the family (gH_e)_g∈Γ/Γ_ξe,e∈P may have intersecting interiors. But as M is geometrically finite and nonelementary, there exists (see [BK,Bow]) t ≥ 0 such that two distinct elements in (gHe[t])_g∈Γ/Γ_ξ

e,e∈P have disjoint interiors. Let tP be the lower bound of all such t’s. For everyγ ∈T¹M , define

maxhtP(γ) =max

e∈Pmaxht_e(γ) and MaxSp(M,P)=maxhtP(T¹M). Remark. LetC be the set of all cusps of M . Under the same hypotheses as in Corol-lary5.4, the following two assertions hold, by applying Corollary5.3to the family of horoballs (gH^′_e^′)_g∈Γ/Γ_ξ

e′ ,e^′∈C with H_e^′′ =He if e^′ =e, and H_e^′′ =H_e^′[t] for some t big enough otherwise, for the first assertion, and to the family (gH_e[tP])_g∈Γ/Γ_ξe ,e∈P

for the second one.

(1) For every cusp e of M , there exists a constant t≥0 such that for every h≥t, there exists a locally geodesic lineγ in M such that maxht_e(γ) =h and maxht_e^′(γ)≤t for every cusp e^′ 6=e in M .

(2) LetP be a nonempty set of cusps of M . Then MaxSp(M,P) contains the halfline [c^′′₂(∞)/2+tP,+∞].

Now, we prove the analogs of Corollaries5.3and5.4for families of balls with disjoint interiors. Let

R^min₀ =7 sinh c^′₁(∞)+3

2c^′₁(∞)≈22.4431.

Corollary 5.5 Let X be a complete simply connected Riemannian manifold with sectional curvature at most−1and dimension at least 3, and let B_n

n∈N be a family of balls in X with disjoint interiors such that the radius R₀ of B₀ is at least R^min₀ . For every h∈

c^′′₁(R^min₀ ,0,0),2R0−2 c^′₁(R^min₀ )

, there exists a geodesic line γ in X with ph_B0(γ)=hand ph_Bn(γ)≤c^′′₂(R^min₀ ) for all n≥1.

Proof. We start by some computations. Letǫ >0. With c1=c^′₁(ǫ) and c₅ =c₅(ǫ,0) as in Subsection4.2, we have c₅≥6 sinh c₁ since c^′₃(ǫ)≥3 by the definition of c^′₃(ǫ) in Equation (- 6 -). By the definition of h0 in Subsection 4.2 and of h^′ in Equation (- 5 -), we have h0(ǫ,0,c^′₁(∞)) ≥h^′(ǫ,sinh c1) ≥2 sinh c1. Hence, by the definitions of c^′′₂(ǫ),c^′′₁(ǫ,0,0),h^′₁ and as ǫ7→c^′₁(ǫ) is decreasing,

c^′′₂(ǫ) =c^′′₁(ǫ,0,0)+c^′₁(∞)+2 c₁

= max{2 c₁,h₀(ǫ,0,c^′₁(∞))+2 c₅}+c^′₁(∞)+2 c₁

≥2 sinh c1+12 sinh c1+c^′₁(∞)+2 c1 ≥14 sinh c^′₁(∞)+3 c^′₁(∞). Define nowǫ=R^min₀ , so that 2ǫ≤c^′′₂(ǫ) and R₀ ≥ǫ. For every n6=0, let Rn be the radius of the ball Bn. If for some n 6=0 we have 2Rn ≤c^′′₂(ǫ), then ph_Bn(γ) ≤c^′′₂(ǫ) and the last assertion of Corollary5.5 holds for this n. Hence up to removing balls, we may assume that Rn ≥ c^′′₂(ǫ)/2 ≥ ǫfor every n 6=0, so that the balls in (Bn)n∈N

areǫ-convex.

The end of the proof is now exactly as the proof of Corollary5.3, with the following modifications: ǫ = R^min₀ ; c^′′₁ = c^′′₁(ǫ,0,0); c^′′₂ = c^′′₂(ǫ); ξ is any point in ∂_∞X ; Cn = Bn for every n in N; we apply Theorem 5.2 instead of Theorem 5.1, which is possible by the range assumption on h; we take now c1 = c^′₁(ǫ), so that we still have d(x^′,y) ≤ c₁ by Proposition 2.3; ξ_n is now the center of B_n, and β_ξn(u,v) = d(u, ξn)−d(v, ξn) (see Section2.1). Besides that, the proof is unchanged.

A heavy computation shows that

c^′₁(R^min₀ )≈1.7627,c^′′₁(R^min₀ ,0,0)≈101.4169 and c^′′₂(R^min₀ )≈106.7051 . Note that the above corollary is nonempty only if R₀ ≥c^′′₁(R^min₀ ,0,0)/2+c^′₁(R^min₀ ) ≈ 52.4712. The constants in the following corollary are not optimal. Theorem1.3in the introduction follows from it.

Corollary 5.6 Let M be a complete Riemannian manifold with sectional curvature at most−1 and dimension at least 3, let (xi)i∈I be a finite or countable family of points in M with r_i =inj_Mx_i, such that d(x_i,x_j)≥r_i+r_j if i6=jand such that r_i0 ≥56 for some i₀ ∈I. Then, for every d ∈[2,r_i0 −54], there exists a locally geodesic line γ passing at distance exactly d from xi0 at time 0, remaining at distance greater than d from x_i0 at any nonzero time, and at distance at least r_i−56 from x_i for every i6=i₀. In particular,

mint∈Rd(γ(t),xi0)=d.

Proof. Let π : Me → M be a universal covering of M , with covering group Γ, and fix a liftex_i of x_i for every i ∈I . Let B_i be the ball B_Me(ex_i,r_i). Apply Corollary5.5to the family of balls (g Bi)g∈Γ,i∈I in X =M , which have pairwise disjoint interiors bye the definition of r_i and the assumption on d(x_i,x_j). Note that r_i0 ≥ 56 ≥ R^min₀ (see the definition of R^min₀ ). Let h =2(ri0 −d), which belongs to [108,2ri0 −4], which is contained in [c^′′₁(R^min₀ ,0,0),2ri0 −2 c^′₁(R^min₀ )] by the previous computations. Then Corollary5.5implies that there exists a geodesic line eγ in M such thate ph_Bi0(eγ) =h and ph_gBi(eγ) ≤ c^′′₂(R^min₀ ) < 108 for all (g,i) 6= (1,i₀). Parametrize eγ such that its closest point to exi0 is at time t = 0. Let γ = π◦eγ, then the result follows by the

definition ofph_C (see Subsection3.1).